Order-distance and other metric-like functions on jointly distributed random variables

We construct a class of real-valued nonnegative binary functions on a set of jointly distributed random variables, which satisfy the triangle inequality and vanish at identical arguments (pseudo-quasi-metrics). These functions are useful in dealing with the problem of selective probabilistic causality encountered in behavioral sciences and in quantum physics. The problem reduces to that of ascertaining the existence of a joint distribution for a set of variables with known distributions of certain subsets of this set. Any violation of the triangle inequality or its consequences by one of our functions when applied to such a set rules out the existence of this joint distribution. We focus on an especially versatile and widely applicable pseudo-quasi-metric called an order-distance and its special case called a classification distance.

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